Question

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let µ denote the true average reflectometer reading for a new type of paint under consideration. A test of H0: µ = 20 versus Ha: µ > 20 will be based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations? (Round your P-values to three decimal places.)(a) n = 16, t = 3.3, a = 0.05P-value =(b) n = 8, t = 1.8, a = 0.01P-value =(c) n = 26,t = -0.6P-value =

Answer 1

Answer:

a) $df=n-1=16-1=15$

The statistic calculated is given by t=3.3  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{15}>3.3)=0.0024$  

So since the p value is lower than the significance level $pv<\alpha$ we reject the null hypothesis.

b) $df=n-1=8-1=7$

The statistic calculated is given by t=1.8  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{7}>1.8)=0.057$  

So since the p value is higher than the significance level $pv>\alpha$ we FAIL to reject the null hypothesis.

c) $df=n-1=26-1=25$

The statistic calculated is given by t=-0.6  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{25}>-0.6)=0.723$  

So since the p value is higher than the significance level $pv>\alpha$ we FAIL to reject the null hypothesis.

Step-by-step explanation:

1) Data given and notation      

$\bar X$ represent the sample mean

$s$ represent the standard deviation for the sample

$n$ sample size      

$\mu_o =20$ represent the value that we want to test    

$\alpha$ represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)      

$p_v$ represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.      

We need to conduct a hypothesis in order to determine if the true mean is higher than 20, the system of hypothesis would be:      

Null hypothesis:$\mu \leq 20$      

Alternative hypothesis:$\mu > 20$      

We don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:      

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$ (1)      

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

(a) n = 16, t = 3.3, a = 0.05, P-value =

First we need to calculate the degrees of freedom given by:  

$df=n-1=16-1=15$

The statistic calculated is given by t=3.3  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{15}>3.3)=0.0024$  

So since the p value is lower than the significance level $pv<\alpha$ we reject the null hypothesis.

(b) n = 8, t = 1.8, a = 0.01, P-value =

First we need to calculate the degrees of freedom given by:  

$df=n-1=8-1=7$

The statistic calculated is given by t=1.8  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{7}>1.8)=0.057$  

So since the p value is higher than the significance level $pv>\alpha$ we FAIL to reject the null hypothesis.

(c) n = 26,t = -0.6, P-value =

 First we need to calculate the degrees of freedom given by:  

$df=n-1=26-1=25$

The statistic calculated is given by t=-0.6  

Since is a one-side upper test the p value would be:      

$p_v =P(t_{25}>-0.6)=0.723$  

So since the p value is higher than the significance level $pv>\alpha$ we FAIL to reject the null hypothesis.